Astronomical Games: August 2002

Figure-Eight in the Sky

A new perspective on an old fascination

To see a world in a grain of sand, /
And a heaven in a wild flower, /
Hold infinity in the palm of your hand, /
And eternity in an hour.

—William Blake, Auguries of Innocence

ONE OF my favorite things to look at when I
was a kid was my dad's globe. This was a National Geographic affair;
it was not mounted, but instead sat freely in a clear plastic stand.
It was also a quality item, and my dad made it clear to me that I was
only to look and touch gently, not throw about like a ball.

I formed all sorts of weird ideas about the globe. It was one of
my first exposures to the idea that we were not on top of the world.
(I grew up in the Bay Area in California.) Instead, we were at a
latitude of 40 degrees, and it occurred to me that we therefore did
not stand up straight, when we thought we were standing up straight.
Instead, we stood at an angle, of 50 degrees to the vertical. If we
had really wanted to stand up straight, we should have leaned over by
an angle of 50 degrees, toward the north. As I said, I formed all
sorts of weird ideas about the globe.

The clear plastic stand, incidentally, had a number of fascinating
symbols and etchings on it. There was a grid of squares, each covering
100 square miles on the globe. There were latitude and longitude
markings, so one could see at an instant how far two different
cities were displaced in those coordinates.

The thing that fascinated me most about the globe, however, was an
unexplained, elongated figure-8 that was unceremoniously placed in the
sparse expanse of the southeast Pacific. What was it, I wondered?
It had the names of the months marked at various points around the
curve, so it clearly had something to do with the year, but what was
the significance? Why was it in the shape of a figure-8? What was it
doing down there in the south Pacific? And couldn't people remember
the months of the year without being reminded by a strange marking on
a globe?

I'm sure you're dying to know the answers to those questions
(well, maybe not the last one), so I'll give them to you, but let me
start the usual way—with something that seems unrelated at first
blush.

In his Republic, Plato (427–347 B.C.) describes—among a
whole host of other things—his curriculum for the ideal schooling in
the Republic. One of the subjects to be studied, as a science, is
that of astronomy.

We must keep in mind, however, that Plato's conception of astronomy
was not what we moderns are used to. The image that most people today
have of astronomers is that of a solitary observer, dwarfed by a
tremendous telescope, staring up at the sky in search of goodness only
knows what. (As a matter of fact, most professional astronomers today
rarely if ever look through the telescopes they use to do their research,
but that's a development of the last century or so.) The job of the
astronomer is to make observations of the heavens, and from those
observations, enhance our knowledge of the cosmos.

That was most certainly not Plato's ideal. His curriculum was
designed in order to form rigorous thinkers, and to that end, the
"real" astronomy was not what was up in the sky. The stars and the
planets showed inconsistencies that were a result of being sensible
objects in the physical world. It would be no more appropriate to
study the "real" astronomy by looking up at the sky than it would be
to study geometry by looking at the imperfect straight lines and round
circles that humans could draw out in the sand. Astronomy was a set
of abstract concepts that could only be approached by logical thought.
(He would surely have been distressed by Hipparchus's attempt to keep
track of the changing heavens by mapping the stars.)

Accordingly, when Plato and his followers sought solutions to
astronomical conundrums, the first criterion by which the solutions
were measured was not how well they matched observations
(although it was something of a consideration of Plato's),
but by how elegant those solutions were. For example, Plato and his
contemporaries felt that the most perfect shape was the circle. It
is as perfectly symmetrical as any shape can be; it is, in a sense,
the figure that all regular polygons aspire vainly to be. So, they
concluded, the ideal astronomical theory for any problem must consist
of circles or combinations of circles.

One such problem was the motions of the planets in the sky. The
planets do not stay in place as the stars do, but instead move
through the constellations. Mostly, they move slowly from west to
east ("prograde" or "direct" motion) as the months pass, but
occasionally, they move east to west ("retrograde" motion). Even such
an idealist as Plato could not ignore that blatant a variation in
motion. After all, the Sun and the Moon don't exhibit
retrograde motion, so there was a clear basis for comparison. But
Plato was no mathematician—he was an idea man, not an analytical
genius. So he was forced to pose this question to others: What theory,
consisting of circles, either in isolation or in combination, could
explain the apparent motion of the planets?

Eventually, a workable solution was arrived at, centuries after
Plato's death, by the Greek astronomer Ptolemy (c. 85–165), in his
geocentric theory of the solar system. But long before Ptolemy, other
Greeks tried their hand at solving Plato's poser. One such person was
Eudoxus of Cnidus (c. 400–347 B.C.), a Greek mathematician and a
contemporary of Plato.

Eudoxus's idea can be imagined as follows. Suppose that you have,
resting on a tabletop, a globe that spins on a tilted axis (unlike my
dad's free-standing globe). Imagine that there's an ant walking along
the equator. Obviously, the ant retraces its path periodically, and we
might call each time around the path one orbit.

Because the globe is tilted, the ant does not stay at the same
height above the table throughout each orbit, but rather rises and
falls. If at one point during its travels, the ant is at its lowest
point, then half an orbit later (and half an orbit earlier as well),
it is at its highest point. Midway between these extremes, the ant
is at its average height.

Now, suppose that instead of putting the globe on a table, you put
it on a turntable, and you set the turntable spinning at exactly the
same rate as the ant's walking, but in the opposite direction. For
example, if we assume that the ant is walking west to east along the
equator—that is, counterclockwise, as seen from above the north
pole—then the turntable is spinning clockwise. Then, because the two
motions roughly cancel each other out, the ant appears to remain more
or less in place (relative to an outside observer).

But not precisely in place. The ant would stay exactly in
place if the globe weren't tilted, for then both the ant and the
turntable would be moving horizontally, and their equal but opposite
rotations would cancel each other out completely. But because the globe
is tilted, the rotations don't cancel out perfectly, and the
ant must at least be sometimes high, sometimes low. After all, without
the turntable, the ant's height goes up and down, and the turntable
can't affect the ant's height; it can only move the ant side to side.

Is that all? Does the ant only move up and down, or
does it trace out a more complex figure? Now, to make that more
precise, suppose you start the globe with the ant on the equator
exactly at its average height, and you shine a laser pointer on the
ant. (It's a weak pointer that doesn't hurt the ant.) As the turntable
rotates clockwise, both the ant and the laser dot move west to east
across the globe, but whereas the ant stays at the same latitude (0
degrees, on the equator), the laser dot appears to change latitude
throughout its orbit. In fact, since the globe is tilted by 23.4
degrees—the tilt of the Earth's axis—the laser dot's latitude
fluctuates between 23.4 degrees north and 23.4 degrees south. Now,
the crucial question: Relative to the laser dot, what is the motion
of the ant—or just as significantly, from the point of view of the ant,
what is the motion of the laser dot?

Eudoxus had sufficient genius for visualization that he arrived at
the surprising but right answer. Here's how he might have reasoned.
If the Earth were flat, you could walk forever in a straight line
without retracing any part of your path. But the Earth is not flat;
instead, as Eudoxus probably suspected, it's a sphere. And since the
sphere is curved, you can't walk a literally straight line. The
curvature of the Earth forces your path to be curved one way or another.
The straightest path you can walk is to go around the Earth in as wide
a circle as possible. One such path is the equator; you can easily see
that by walking along the equator, you are neither turning north nor
south. Another way to walk as straight as possible is to start at the
north pole, walk due south along some particular line of longitude until
you get to the south pole, and then return to the north pole along the
"opposite" line of longitude.

Each of these straightest paths is called a great circle.
There are an infinite number of them on the Earth, or on the globe, or
indeed on any sphere. Each of them has the same diameter as the sphere,
and the center of any great circle is the same as the center of the
sphere. The ant on the globe traces out a great circle—namely, the
equator. The laser dot traces out another great circle, but one that
is horizontal and therefore not the equator. Since the globe
is tilted by 23.4 degrees, the laser dot's great circle is tilted to
the equator by 23.4 degrees as well. These two circles intersect at
two opposite points, which must obviously be along the equator, 180
degrees apart. This is the key to Eudoxus's idea.

Suppose we start with the ant and the laser dot at the same spot
again. The ant proceeds directly eastward along the equator. The
laser dot follows a great circle that is inclined to the equator, by
23.4 degrees, either to the northeast, or the southeast. For the
sake of discussion, let's suppose that the laser dot is moving to
the northeast of the original starting point.

At first, the ant and the laser dot are still close together, and
we can for all practical purposes ignore the spherical shape of the
globe, just as, in real life, we can ignore the spherical shape of the
Earth when navigating inside our home. Since the ant and the laser
dot are moving at the same speed, they appear to be carried along at
the edge of an ever-expanding compass dial, as in Figure 1.

Figure 1. The beginning of the ant and dot's paths.

Initially, the laser dot seems to be moving mostly northward,
relative to the ant. But because the ant puts all of its motion into
the eastward direction, and the laser dot only puts most of it there,
the laser dot must also appear to be moving slightly westward, from
the standpoint of the ant. (See Figure 2.)

Figure 2. The beginning of the dot's path, relative to the ant.

If the globe were actually flat, the ant and laser dot would spread
out forever, with the dot always moving to the north-northwest of the
ant. But the globe isn't flat, and if the ant and laser dot continue
far enough, the globe's curvature will come into play.

For example, after a quarter of an orbit, the ant is 90 degrees
(1/4 of 360) away from its starting point, along the equator. The
laser dot, travelling at the same rate, is also 90 degrees from its
starting point, but north of the ant. You might expect that it would
also be somewhat to the west of the ant, as before, but it's not.
Instead, it's exactly due north of the ant. (See Figure 3.)

Figure 3. The dot's path, relative to the ant, after a quarter orbit.

What has happened? The new factor is that the laser dot's path is
taking it to higher latitudes on the globe, where the lines of longitude
are closer together. As they both approach the 1/4-orbit point in their
travels, therefore, the laser dot is gaining on the ant in longitude.
This makes up perfectly for the start of their voyages, where the ant
moved out ahead of the dot in longitude, so by the time that they have
gone through a quarter orbit, both the laser dot and the ant have moved
through exactly 90 degrees of longitude.

If we follow their motion further, into the second quarter of the
orbits, the laser dot now races ahead of the ant in longitude. But we
know that they must meet again after both have travelled through a half
orbit; at that time, they must both be on the opposite side of the globe
from their original starting point. As seen in Figure 4, from the point
of view of the ant, the laser dot must have travelled in a wide looping
path, starting toward the north-northwest, then curving eastward, then
returning from the north-northeast.

Figure 4. The dot's path, relative to the ant, after a half orbit.

In the second half of their orbits, the exact same thing happens,
except inverted. Again, the laser dot, with some of its motion toward
the south, falls behind the ant in longitude, and it appears to the ant
to be moving to the south-southwest. Then, as it moves to more
southern latitudes, where the lines of longitude are closer together,
it catches up with and overtakes the ant in longitude. Finally, as its
path takes it back toward the equator, the ant and the laser dot meet
once more at the starting point, one orbit later for each. (See Figure
5.)

Figure 5. The dot's path, relative to the ant, after a complete orbit.

This figure-8 shape is the path that the laser dot appears to take
from the perspective of the ant. The amazing thing is that Eudoxus was
able to figure this all out without the benefit of actual globes or laser
pointers. To him, incidentally, the looping path, retracing itself
over and over again, resembled the loops placed around a horse's feet
to fetter it, so he called the path a "horsefetter." Naturally, he
spoke Greek, so the word he used was hippopede, pronounced
"hip-POP-puh-dee," from the Greek words for "horse" and "feet."

Eudoxus thought that by superimposing this figure-8 loop on a third,
underlying west-to-east motion, he could simulate the retrograde motion
of the planets. Half the time, the hippopede would also be moving
west to east, so the combined motion would be west to east as well—this
would be prograde, or direct, motion. Even much of the rest of the
time, the hippopede would not be moving enough in the opposite direction
to counteract the general west-to-east translation. Only when the
hippopede was moving nearly as fast as possible, east to west, would
there be a resulting backward slide, and this backward slide Eudoxus
identified as retrograde motion.

It was a clever bit of explanation, but there were a number of
problems with it. First of all, if it were correct, then all of the
retrograde loops should have been symmetrical, and that wasn't so.
Secondly, and more seriously, all the planets should remain at the
same brightness throughout their orbits, and they certainly did not.
Mars, in particular, is dozens of times brighter at some times than
at others. For these reasons, Eudoxus's hippopede was eventually
replaced, first by Ptolemy's theory of deferents and epicycles,
equants and eccentrics, and 1,400 years thereafter by Copernicus and
the heliocentric theory.

The hippopede re-entered science, though, in a completely
unexpected way—a way that was only opened up by the advent of accurate
timekeeping.

For millennia, humans kept track of time by noting the general
location of the Sun. One might speak of leaving for town at sunrise,
or of returning when the Sun was a hand's breadth above the horizon,
and so forth. The Sun's motion was sufficiently constant to provide
a convenient basis for telling time.

At some point, it became expedient to divide both the day and the
night into portions, and the Babylonians chose to divide them both
into 12 equal parts called "hours," from an ancient Greek word meaning
"time of day." Twelve was a useful number, in that a quarter, or a
third, or a half of a day or night all came out to a whole number of
hours. These hours could be labelled on a sundial, so the moving
shadow of a stylus, or gnomon, would mark out the advancing hours—at
least, during the daytime.

Unfortunately, all of the daytime hours were equal to each other,
and all of the nighttime hours were also equal, but the daytime hours
were not the same length as the nighttime hours. Instead, they were
longer in summer (naturally) and shorter in winter. The explanation
for this was in the changing height of the Sun. It rose higher in the
sky in summer, and more of its circular path was then above the horizon,
so naturally the 12 daytime hours took longer to pass. In the winter,
exactly the opposite was true: the Sun did not get very high at all
in the sky, even at its peak. Less of its circular path was above the
horizon, so the 12 daytime hours took less time to pass.

Eventually, other devices for telling time were developed that did
not depend on the slightly variable nature of the Sun's path: for
instance, hourglasses, or burning candles. With the introduction of
these timekeepers, the variations in the daytime and nighttime hours
became quite troublesome. It was tedious to have to change candles or
hourglasses with each month. How much easier it would be to replace
the inconstant hours with 24 equal ones. The only inconvenience was
that sunrise and sunset would take place at slightly different hours
throughout the year, but that could easily be accounted for.

Then, in 1656, the Dutch astronomer and physicist Christiaan Huygens
(1629–1695) developed the first pendulum clock. Galileo had had the
idea previously, while watching a chandelier sway back and forth in a
cathedral, but had never followed through on a design. Huygens was
the first to overcome the physical obstacles to building a clock
based on the principle of the pendulum, and he ushered in the era of
precision timekeeping.

Huygens's clock was also the first to be accurate to minutes a
day, and the clock face gained another hand. Later clocks were even
accurate to seconds, and now was discovered an interesting discrepancy.
The moment that the Sun crosses the meridian—an imaginary north-south
line in the sky—is called local noon, after an old word meaning the
ninth (daytime) hour of the day. (This was midafternoon, but later
was moved back earlier, to midday.) By all rights, the time between
local noon on two successive days should be exactly 24 hours. But
as measured by these accurate clocks, the interval between two
consecutive local noons was sometimes a few seconds long; at others,
a few seconds short. If we set a clock exactly to noon when the Sun was
at local noon on one day, then the next day, the Sun would reach local
noon, not at 12:00 exactly, but perhaps at 11:59:58, or at 12:00:10.
These discrepancies added up, so that at various times of the year,
the Sun was as much as a quarter of an hour "early" or "late." The
errors repeated in a cycle of length one year, year after year.

Either the clocks were wrong, or the Sun's apparent motion across
the sky was not as constant as previously thought (or both). We now
know that it's the latter, and this repeating cycle is called "the
equation of time" by astronomers. The Sun does not go at the same
rate in right ascension (the astronomical version of longitude) all
year long, but instead moves through lines of right ascension faster
at some times, slower at others. At no point does it actually go
the "wrong" way—it doesn't exhibit retrograde motion, in other
words—but this variation is what causes the Sun to cross the meridian
early or late. And if we plot the "location" of the Sun, with its
northern and southern advances drawn along the vertical axis, and its
earliness or lateness drawn along the horizontal axis, we get the
figure drawn on my dad's globe, which is called an "analemma." (See
Figure 6.)

Figure 6. The analemma.

The word "analemma" is Greek for the pedestal of a sundial, and
itself comes from the Greek verb analambanein, meaning "to
take up, to resume, to repair," so that the pedestal is something that
supports the sundial upon it. Early on, "analemma" seems to have been
extended to refer to a particular kind of sundial, in which only the
height of the Sun was indicated, by measuring the size of the shadow
cast by the sundial. Later, it was used for a number of meanings
related to the height of the Sun; its latest meaning, and that with
which we are interested here, is some kind of representation of the
Sun's gradually changing path in the sky at the same time (noon by
the clocks) each day.

It surely hasn't escaped your attention that the analemma and
Eudoxus's hippopede share a certain resemblance, a resemblance that,
as it turns out, is more than accidental. The hippopede results from
the conjunction of two circular motions, and so does the analemma.

The apparent motion of the Sun is really due to two motions
of the Earth. One is the Earth's orbit around the Sun. The
Earth completes one revolution about the Sun in one year, and if that
were the only motion that the Earth had, then we on the Earth would
see the Sun appear to go around the Earth just once a year.

However, the Earth has a second motion: its rotation on its axis.
It does so approximately once a day, and it is for that reason, mostly,
that the Sun appears to revolve around the Earth once each day.
Since these two motions have periods in approximately the ratio
365.25:1 (the number of days in a year), while the hippopede results
from two motions with equal periods, you might think that the hippopede
doesn't have much relevance to the analemma.

But you'd be wrong. As I mentioned, the Earth rotates on its
axis only approximately once a day, and the Sun's apparent
motion across the sky is only mostly due to this rotation.
A tiny component is due to the first motion of the Earth, its orbital
revolution. Since this revolution takes 365.25 times longer than the
rotation, it contributes 1/365.25 as much to the Sun's apparent motion
across the sky as does the Earth's rotation. Now, the Earth's rotation
makes the Sun seem to move east to west, from dawn to dusk, but its
orbital revolution appears to add a second component, from west to east.
This second component very slightly counteracts the first, so that the
24-hour day is longer than you might expect based solely on rotation.
In fact, the Earth actually rotates on its axis, with respect to the
stars, every 23 hours, 56 minutes, and 3.5 seconds. This slightly
shorter day is called the "sidereal day," after a Latin word meaning
"star," since this is the time it takes for the Earth to rotate once
relative to the stars. The extra four minutes each day is due to the
Earth's orbit around the Sun, and is 1/365.25 of the 24-hour day.

In other words, if the Earth didn't revolve around the Sun, but
only rotated in place, in defiance of the law of gravity, the Sun would
appear to go once around the Earth in 23 hours, 56 minutes, and 3.5
seconds, instead of the customary 24 hours. And if we were to take a
snapshot of the Sun every day at the same time by the clock, it
would be 3 minutes and 56.5 seconds further along each day. After
two days, it would be ahead (that is, further west) by 7 minutes and
53 seconds; after three days, by 11 minutes and 49.5 seconds; after
four days, by 15 minutes and 46 seconds, and so forth.

How long would it take for this margin to extend to 24 hours, so
that the Sun would once again be "on time," on the meridian at noon?
Why, as many times as 3 minutes and 56.5 seconds goes into 24
hours—and as we noted above, this interval is 1/365.25 of 24 hours,
so it would take 365.25 days for the Sun to "lap" the 24-hour clock.
A year, in other words. In short, if the Earth only rotated, and
didn't revolve around the Sun, the Sun would appear to revolve
around us every 23 hours, 56 minutes, and 3.5 seconds, but by taking
snapshots of the Sun every 24 hours, which is just about four minutes
longer, this motion would appear to be slowed down to just one
revolution per year.

In case that sounds confusing, it's like watching a car drive by
you on the road. In reality, the car's wheels may be rotating very
rapidly—let's say, 25 times a second. (That'd be one fast car, by the
way—probably around 150 to 200 kilometers an hour!) But if you watch
a film of the car, where the camera takes 24 frames per second, each
frame catches the wheel when it has gone through 1-1/24 of a rotation.
Since the eye can't tell the difference between 1-1/24 of a rotation
and just 1/24 of a rotation, it appears as though the wheel is actually
rotating at only 1/24 rotation per frame. That works out to one rotation
every 24 frames—or once a second.

In much the same way, when we take our figurative snapshots of
the Sun every 24 hours, the Earth's rotation, alone, makes the Sun appear
to revolve around the Earth, once a year, from east to west, along a path
called the celestial equator. Meanwhile, as described above,
the Earth's orbital revolution, alone, makes the Sun appear to revolve
around the Earth, once a year in the opposite direction, from west to
east, along another path called the ecliptic. Both the
celestial equator and the ecliptic are great circles. What's more, these
two great circles are not the same, but because of the Earth's axial tilt,
are instead inclined to one another by an angle of 23.4 degrees.

We therefore have an exact analogue of Eudoxus's hippopede, but
this time applied to the apparent motion of the Sun throughout the year.
These two motions combine to create the figure-8 shape of the analemma.
Eudoxus could not possibly have known about this application of his
theory, which was originally designed to account for the retrograde
motion of the planets. As an explanation of that behavior,
the hippopede was basically dead on arrival. Too bad that accurate
clocks were not available in his day; otherwise, he might have found
the right use for his geometric intuition.

But one last objection remains: The analemma on the globe is not a
symmetric figure-8 at all! Rather, it's smaller on the northern end,
and larger on the southern end. Why is that?

That asymmetry is due to one further property of the Earth's orbit
around the Sun: its eccentricity. The Earth's orbit is nearly circular,
but not precisely so. It is actually an ellipse, and the Earth moves
along that ellipse in accordance to Kepler's laws of planetary motion.
(See "Music of the Ellipses.") As such,
the Earth moves faster when it is closer to the Sun, and slower when it
is further from the Sun, and this translates to a corresponding variation
in the Sun's apparent west-to-east motion due to the Earth's revolution.
Just how elliptical the orbit is, and the angle between the
long axis of the orbit and the axis of the Earth, determine the contour
of the analemma.

Incidentally, I'm not certain just why the analemma is specifically
in the southern Pacific—perhaps because that's the least crowded part
of the planet, cartographically speaking—or why it's needed on a globe
at all. It does have some significance to sundial builders, since it
can be used to correct for the equation of time, if the months of the
year are marked out (as they are on my dad's globe) and one rotates the
dial of the sundial according to the analemma. But it doesn't seem to
need to be on a globe, and indeed, more modern globes now eschew the
analemma in favor of a more extensive legend.

Here is a C program to
compute and plot the analemma for various different orbital parameters.
It's not tremendously user friendly, and can probably use some additional
documentation. (It also uses the "system" call, which probably should
be replaced with something in the "exec" family, if that means anything
to you.) However, it uses the ideas presented in this essay,
with the additional amendment that the eastward march due to the Earth's
orbital revolution varies in speed because that orbit is elliptical.
This approach is more accurate than programs where the effects on the
equation of time of the two motions is added linearly (see, for example,
http://www.analemma.com/). That's
reasonably accurate for small eccentricities and axial inclinations, but
becomes noticeably inaccurate for extreme orbits.